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A Generalization of the Turán Theorem and Its Applications

Published online by Cambridge University Press:  20 November 2018

Yu-Ru Liu
Yu-Ru Liu*
Affiliation:
Department of Pure Mathematics University of Waterloo Waterloo, ON N2L 3G1, e-mail: yrliu@math.uwaterloo.ca
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Abstract

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We axiomatize the main properties of the classical Turán Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field.

Keywords

11N3711N80
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 47 , Issue 4 , 01 December 2004 , pp. 573 - 588
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Elliott, P. D. T. A., Probabilistic number theory, I, II. Springer-Verlag, New York, 1979.Google Scholar
[2] Hartshorne, R., Algebraic geometry. Springer Verlag, New York, 1977.Google Scholar
[3] Hardy, G. H. and Ramanujan, S., The normal number of prime factors of number n. Quar. J. Pure Appl. Math. 48 (1917), 76–97 Google Scholar
[4] Liu, Y.-R. and Murty, M. R., The Turán sieve method and some of its applications. J. RamanujanMath. Soc. 14 (1999), 21–35.Google Scholar
[5] Lorenzini, D., An invitation to arithmetic geometry. Graduate Studies in Mathematics, 9, American Mathematics Society, Providence, RI, 1996.Google Scholar
[6] Lang, S. & Weil, A., Number of points of varieties in finite fields. Am. J. Math. 76 (1954), 819–827.Google Scholar
[7] Mertens, F., Ein Beitrag zur analytischen Zahlentheorie. J. Reine Angew.Math. 78 (1874), 46–62.Google Scholar
[8] Murty, M. R., Problems in analytic number theory. Springer-Verlag, New York, 2001.Google Scholar
[9] Prachar, K., Verallgemeinerung eines satzes von Hardy and Ramanujan auf algebraische zahlkörper. Monatsh.Math. 56 (1952), 229–232.Google Scholar
[10] Rosen, M., Number theory in Function Fields. Lecture notes given at K.A.I.S.T., Taejon, Korea, 1994.Google Scholar
[11] Saidak, F., On a theorem of Hardy-Ramanujan-Turán I, to appear.Google Scholar
[12] Turán, P., On a theorem of Hardy and Ramanujan. J. LondonMath. Soc. 9 (1934), 274–276.Google Scholar
[13] Weber, H., Über zahlengruppen in algebraischen körpern. Math. Ann. 49 (1897), 83–100.Google Scholar